In this paper the Poincaré–Hopf inequalities are shown to be necessary and sufficient conditions for an abstract Lyapunov graph L to be continued to an abstract Lyapunov graph of Morse type. The Lyapunov graphs considered may represent smooth flows on closed orientable n-manifolds, n \geq 2. The continuation which is presented by means of a constructive algorithm, is shown to be unique in dimensions two and three. In all other dimensions, the exact number of possible continuations of L are presented.

In this paper we prove that on an Anosov manifold the space of symmetric m-tensor fields of vanishing energy is finite dimensional modulo the space of potential tensor fields for an arbitrary m and coincides with the latter for m=0 and m=1. For m=2 this question relates to the spectral rigidity problem.

The exact order of mixing for zero-dimensional algebraic dynamical systems is not entirely understood. Here non-Archimedean norms in function fields of positive characteristic are used to exhibit an asymptotic shape in non-mixing sequences for algebraic \mathbb{Z}^2-actions. This gives a relationship between the order of mixing and the convex hull of the defining polynomial. Using these methods, we show that an algebraic dynamical system for which any shape of cardinality three is mixing, is mixing of order three, and for any k\geq 1 exhibit examples that are k-fold mixing but not (k+1)-fold mixing.

Hölder continuous observations on hyperbolic basic sets satisfy strong statistical properties such as exponential decay of correlations, central limit theorems and invariance principles (approximation by Brownian motion).

Using an equivariant version of the Ruelle transfer operator studied by Parry and Pollicott, we obtain similar results for equivariant observations on compact group extensions of hyperbolic basic sets.

In this paper we consider an analog of the regions of instability for twist maps in the context of area preserving diffeomorphisms which are not twist maps. Several properties analogous to those of classical regions of instability are proved.

A dynamical 2-complex is a 2-complex equipped with a set of combinatorial properties which allow one to define non-singular semi-flows on the complex. After giving a combinatorial characterization of the dynamical 2-complexes which define hyperbolic attractors when embedded in compact 3-manifolds, we give an effective criterion for the existence of cross sections to the semi-flows on these 2-complexes. In the embedded case, this gives an effective criterion of existence of cross sections to the associated hyperbolic attractors. We present a similar criterion for boundary-tangent flows on compact 3-manifolds which are constructed by means of our dynamical 2-complexes.

We study Ruelle's transfer operator \mathcal{L} induced by a C^{\mathbf{r}+1}-smooth expanding map \varphi of a smooth manifold and a C^{\mathbf{r}}-smooth bundle automorphism \Phi of a real vector bundle \mathcal{E}. We prove the following exact formula for the essential spectral radius of \mathcal{L} on the space C^{\mathbf{r},\alpha} of \mathbf{r}-times continuously differentiable sections of \mathcal{E} with \alpha-Hölder \mathbf{r}th derivative:

{\rm r}_{\rm ess}(\mathcal{L};C^{\mathbf{r},\alpha})=\exp\Big(\sup_{\nu\in{\rm Erg}} \{h_\nu+\lambda_\nu-(\mathbf{r}+\alpha)\chi_\nu\}\Big),

where Erg is the set of \varphi-ergodic measures, h_\nu the entropy of \varphi with respect to \nu, \lambda_\nu the largest Lyapunov exponent of the cocycle induced by \Phi, and \chi_\nu the smallest Lyapunov exponent for the differential D\varphi.

Nous étendons des théorèmes sur le spectre de Markoff et de Lagrange connus dans le cas rationnel au cas analogues imaginaires quadratiques. Pour ce faire, nous nous ramenons à un problème de géométrie hyperbolique.

Given an irreducible subshift of finite type X, a subshift Y, a factor map \pi:X\to Y, and an ergodic invariant measure \nu on Y, there can exist more than one ergodic measure on X which projects to \nu and has maximal entropy among all measures in the fiber. However, there is an explicit bound on the number of such maximal entropy preimages.

We consider a class of billiard tables (X,g), where X is a smooth compact manifold of dimension two with smooth boundary \partial X and g is a smooth Riemannian metric on X, the billiard flow of which is completely integrable. The billiard table (X,g) is defined by means of a special double cover with two branched points and it admits a group of isometries G \cong \mathbb{Z}_2 \times\mathbb{Z}_2. Its boundary can be characterized by the string property; namely, the sum of distances from any point of \partial X to the branched points is constant. We provide examples of such billiard tables in the plane (elliptical regions), on the sphere \mathbf{S}^2,on the hyperbolic space \mathbf{H}^2, and on quadrics. The main result is that the spectrum of the corresponding Laplace–Beltrami operator with Robin boundary conditions involving a smooth function K on \partial X uniquely determines the function K, provided that K is invariant under the action of G.

Soient G un groupe de Lie semi-simple de centre fini et \Gamma un sous-groupe discret Zariski dense de G. Dans des travaux précédents, nous avons défini l'indicateur de croissance de \Gamma, objet qui généralise en rang supérieur l'exposant critique considéré en \mathbb{R}-rang 1, et nous avons associé à \Gamma des généralisations des mesures de Patterson-Sullivan. Dans cet article, nous donnons des précisions sur ces objets quand \Gamma est un sous-groupe du type de Schottky, en les reliant au formalisme thermodynamique.

We prove that the solenoid with two different contraction coefficients has zero Hausdorff and positive packing measure in its own dimension and the SBR measure is equivalent to the packing measure on the attractor. Further, we prove similar statements for Slanting Baker maps with intersecting cylinders (in \mathbb{R}^{2}).

We investigate properties of a class of martingales formed by picking a measurable set A in a compact group G, taking random rotates of A, and taking measures of the resulting intersections, suitably normalized. Such martingales are shown to yield measures on G^\infty that are singular with respect to product Haar measure, and their ergodic decompositions with respect to the shift map are analyzed. Connections are made with De Finetti theory.

We prove that fairly general spaces of tilings of \mathbb{R}^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in the second author's recent work, and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a \mathbb{Z}^d subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that (1) the tiles are assumed to be polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to full-face), (2) only a finite number of tile types are allowed, and (3) each tile type appears in only a finite number of orientations. The proof is constructive and we illustrate it by constructing a ‘square’ version of the Penrose tiling system.

For a continuous transformation f of a compact metric space (X,d) and any continuous function \phi on X we consider sets of the form

K_{\alpha} =\bigg\{x\in X:\lim_{n\to\infty} \frac 1n \sum_{i=0}^{n-1} \phi( f^i(x))=\alpha \bigg\},\quad\alpha\in\R.

For transformations satisfying the specification property we prove the following Variational Principle

h_{\rm top}(f,K_{\alpha}) = \sup\bigg( h_\mu(f): \mu\text{ is invariant and } \int\phi \,d\mu=\alpha \bigg),

where h_{\rm top}(f,\cdot) is the topological entropy of non-compact sets. Using this result we are able to obtain a complete description of the multifractal spectrum for Lyapunov exponents of the so-called Manneville–Pomeau map, which is an interval map with an indifferent fixed point.

We also consider multi-dimensional multifractal spectra and establish a contraction principle.